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0593_C11_fm Page 387 Monday, May 6, 2002 2:59 PM

Generalized Dynamics: Kinematics and Kinetics 387

From Figure 11.10.7, we see that N , N , and N may be expressed in terms of the unit
1
3
2
vectors n , n , and n as:
3
1
2
N = cosφ n − sin cosθ n +sin sinφ n
φ
θ
1 1 1 3
φ
φ
N = sinφ n + cos cosθ n − cos sinθ n (11.10.40)
2 1 2 3
N =sinθ n +cosθ n
3 2 3
Hence, by substituting into Eq. (11.10.39) v becomes:
G
v = ( ψφ ˙ n ) θ − rθ ˙ n (11.10.41)
r ˙
+ sin
G 1 2
Also, the angular velocity ωω ωω of D relative to S may be expressed as (see Eq. (4.12.2)):
)
˙
˙
˙
φ
φ
ωω= θn +( ψ ˙ + sinθ n + cosθn (11.10.42)
1 2 3
Observe that the result of Eq. (11.10.41) could have been obtained directly as ωω ωω × rn , the
3
expression for velocities of points of rolling bodies (see Section 4.11, Eq. (4.11.5)).
The partial velocities of G and the partial angular velocities of D with respect to θ, φ,
and ψ are then:
v =− r n , v = r sinθ n , v = r n 1 (11.10.43)
G
G
G
˙ ψ
˙ φ
1
2
˙ θ
and
ωω = n 1 , ωω = sinθn 2 + cosθn 3 , ωω = n 2 (11.10.44)
˙ ψ
˙ θ
˙ φ
*
The inertia force system on D may be represented by a single force F passing through G
*
*
*
together with a couple with torque T where F and T are (see Section 8.13, Eqs. (8.13.7)
to (8.13.12)):
G
*
m
F =−m a =− (a n + a n + a n ) (11.10.45)
1 1 2 2 3 3
and
*
T = T n + T n + T n (11.10.46)
1 1 2 2 3 3
where
T =−α Ι + ω ω − Ι 33)
2 (Ι
1 1 11 3 22
T =−α Ι + ω ω − Ι 11) (11.10.47)
3 (Ι
2 2 22 1 33
1 (Ι
T =−α Ι + ω ω − Ι 22)
3 3 33 2 11